[Task]: Write integration tests to verify that prompt caching works

[kantord]

Description

blocked by #892

Write integration tests to ensure prompt caching works with CodeGate and prevent regressions

We know from this github issue that prompt caching is currently broken with Anthropic thru CodeGate — once this is fixed, we should write integration tests around this.

We should also be able to do the same for Anthropic via OpenRouter.

Additional Context

Here is a document explaining everything we know about prompt caching. Probably most of this document won't be relevant for the integration testing task, but if you feel lost, it could be a good starting point.

https://docs.google.com/document/d/1gEgkIyT1CRcTMAwYiU6ldeeosKKbWMX5HM8AdQx3xV0/edit?tab=t.0